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Now if we have an isotropic point light source of power P,

Number of photons $$N=\frac{P}{E} = \frac{P \lambda}{hc}$$

Hence one can find the change in momentum and hence the force exerted by a beam or light sources.

But lets say we keep an isotropic light source at a point on the optical axis of a convex lens such that rays emerge parallel to axis after refraction.The lens does not absorb nor reflect any light energy.

Then the force exerted by the light source on lens is given by:

$$F=\frac{P}{2c}(1-\frac{d}{\sqrt [2]{r^2 + d^2}} - \frac{r^2}{2(r^2 + d^2)})$$

How do derive the expression?

My attempt:

Lets say a ray approaches at an angle ##\theta##

Then initial momentum of a single photon = ##m(vcos\theta + vsin\theta )## and final momentum is ##mv##

Hence, $$\delta p = \frac {h}{\lambda}(1-cos\theta -sin\theta)$$